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Chaotic Dynamics

Title:ANOMALOUS SCALING OF THE PASSIVE SCALAR

Abstract: We establish anomalous inertial range scaling of structure functions for a
model of advection of a passive scalar by a random velocity field. The velocity
statistics is taken gaussian with decorrelation in time and velocity
differences scaling as $|x|^{κ/2}$ in space, with $0\leqκ< 2$. The
scalar is driven by a gaussian forcing acting on spatial scale $L$ and
decorrelated in time. The structure functions for the scalar are well defined
as the diffusivity is taken to zero and acquire anomalous scaling behavior for
large pumping scales $L$. The anomalous exponent is calculated explicitly for
the $4^{\m\rm th}$ structure function and for small $κ$ and it differs
from previous predictions. For all but the second structure functions the
anomalous exponents are nonvanishing.